VECTORS

Vectors A person walks 5 meters South,

then 6 meters West.

How far did he walk?

Vectors A person walks 5 meters North,

then 6 meters East.

How far did he walk?

What is his displacement?

Scalars & Vectors1. A scalar is a quantity that has

magnitude (size) only.2. Vectors are quantities that have

size and direction.

Examples Scalars: 5 m. 7 m/s.

Vectors: 5 m South. 7 m/s up.

Vectors Vectors have “magnitude”

5 m 7 m/s

AND

Vectors have “direction”South Up

Drawing vectors. Vectors are drawn as arrows.

Length of the arrow is proportional to the magnitude of the vector.

(The bigger the number, the longer the arrow!)

Vector Arithmetic Vectors can be multiplied by

numbers.

If A = 5 m North, 2A = 10 m North

Vector Arithmetic Negative signs flip the direction.

Vectors1. Vectors can be added together.2. To show vector addition graphically,

you need to draw the vectors:1. To scale2. “Tip to Tail”.

The red line is the sum of the other two vectors and is called the resultant.

Adding vectors gives a displacement, not a distance.

Graphic Addition Examples

Switch to Notebook to draw vector answers or write on White Board on top of projection.

Vector Arithmetic

Common notation A vector will usually be given as a

length “at” some angle.

I.e. 4 m @ 60◦ or 7 m @ 235◦

Common Notation The angle listed will be measured

counter clockwise from the + x axis.(unless otherwise specified)

Common Notation The angle listed will be measured

counter clockwise from the + x axis.(unless otherwise specified)

ADDING VECTORS ALGEBRAICALLY

1. Adding vectors using algebra gives us a much more exact number than doing it graphically.

2. To add vectors algebraically, you need to remember a little geometry and trigonometry.

PERPENDICULAR VECTORS

1. When vectors are at right angles to each other, we can find both the size and direction.

a. We find the magnitude of the resultant by using the Pythagorean Theorem (a2 + b2 = c2 ).

b. We find the direction (angle) of the resultant by using inverse cosine or inverse tangent functions.

Find the resultant. A= 6 m @ 0 deg.

B = 9 m @ 90 deg.

What is A + B?

Adding parts. Add parts together to get same answer

as adding wholes together, just like adding vectors. For example:

Adding Wholes: 2 + 7 = 9 Adding Parts: 2 = 1 + 1 and 7 = 3 + 4

so (1 + 1) + (3 + 4) = 9 and is the same

as adding wholes above.

Components

Any vector can be treated like the sum of 2 vectors or broken into 2 parts.

A vector can be treated like a horizontal (x) vector and a

vertical (y) vector added together.

X means a unit vector in the X direction.y means a unit vector in the y direction.

Components Vector A

Ax is the “horizontal component” of A

Ay is the “vertical component” of A

Trig Review Vector A of length A is the hypotenuse. X is the length or magnitude of the X component (vector Ax). Y is the length or magnitude of the Y component (vector Ay). Cos = (X/A) Sin = (Y/A) or X = A Cos Y = A Sin Inverse Cos is written as Cos -1 . Inverse Cos asks what angle produces that Cos. Cos -1 (X/A) gives or Tan -1 (Y/X) gives

= X

= y

ADDING NON-PERPENDICULAR

VECTORS1. When vectors are applied at different

angles, we need to break them into their vertical (y) and horizontal (x) pieces.

2. Then we can find (Σ is sigma = sum)a. Σ x and Σ y b. Use Pythagorean Theorem to find

resultant magnitude (M). (Σ x)2 + (Σ y)2 = M2

c. Use Inv Cos = Σ x to find resultant M direction

angle.

EXAMPLE

A = 50 km at 110 degB = 30 km at 60 deg

= 60o = 110o

A = 50 km B = 30 km

Example We want A + B

EXAMPLE

The resultant is the vector made from the combined components of the two original vectors.

θ = 60o

θ = 110o

A = 50 km B = 30 km

X = Ax + BX “The Horizontal of A + The horizontal of B”

X = 50*cos(110) + 30 *cos(60)

X = -17.1 + 15 = -2.1

Y = AY + BY “The Vertical of A + The Vertical of B”

Y = 50*sin(110) + 30 *sin(60)

Y = 47.0+ 26.0 = 73.0

C = -2.1 Horizontal

73 Vertical

How long is C?

C2 = (-2.1)2 + (73)2

C = 73.03

What is the angle of C?

cos cos -

1(

degrees

Find the resultant C = 5 km at 30 deg. D = 9 km at 45 deg.

Find C + D